If you have not read my previous post on triangle congruence, make sure you do. Anyway back to the angle-side-angle and angle-angle-side theorems. I will prove even more triangles congruent in this post! Like I wrote before, that is all I really remembering doing in high school geometry.

Example 1 Triangle Congruence Proof with AAS

The first example is example 3 in the video. If you are wondering why I am writing about something I already did in a video, let me explain. It has to do with different learning styles. It is pretty important to understand your learning style. I will have to blog about learning styles soon. Anyway, through my blog, I try to reach a variety of learning styles. Thus, I create graphics, write and create videos to get my math point across. Anyway, on with example 3 from the video.

When I look at what is given and the diagram, I notice the two triangles share a side which will be a corresponding congruent side, segment PR. Since, segment RP bisects angle SRQ, that means angles SRP and QRS are congruent corresponding parts by the definition of angle bisector. Those two facts taken with the given, provides enough proof to establish AAS in both triangles SRT and QRP.

Example 2 Triangle congruence with AAS

Given: angles B and D are congruent and segment AB is parallel to segment CD

Prove: triangles ABC and CDA are congruent

As is true with any proof, you need to understand the given and how it will help you identify a pair of correspond angles or sides of a triangle congruent to use one of the methods to proving triangles congruent. From the given, we have a pair of corresponding angles congruent, D and B. From the diagram a pair of corresponding sides can be established. AC is a shared side and with the reflexive property of congruence it can stated AC is congruent to AC. Also in the given, it is stated that segments AB and DC are parallel. Whenever you hear the words parallel lines, you must remember the special angle pairs formed by two parallel lines and a transversal. In this case, angle BAC is congruent to angle DCA because of the alternate interior angles theorem. With all of that, it can be said that triangle ABC is congruent to triangle CDA by the AAS theorem.

Example 3 Proving Triangles Congruent with AAS

Given: Segments XQ and TR are parallel and segment XR bisects QT

Prove: triangle XMQ congruent to triangle RMT

This proof requires the most work of all the proofs I have done on this blog. There are no corresponding parts given as congruent, which means we have to establish three pairs of corresponding parts congruent. The two keywords in the given are parallel and bisects. Angle X and angle R are alternate interior angles and are congruent because the two angles are formed by two parallel lines and a transversal. Angles XMQ and RMT are congruent because all vertical angles are congruent. I have two angles and need to prove 1 pair of corresponding sides congruent. Those sides will be segment TM and segment QM by the definition of segment bisector. Since the sides are the non-included sides, triangle XMQ is congruent to triangle RMT by the AAS theorem.

In the video, it is not labeled example 1, but the first bit of information is critical to this lesson.

Angle Side Angle Postulate

It two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Angle Angle Side Theorem

It two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of another triangle, then the two triangles are congruent.

It would be good to remember that a postulate is something that is assumed to be true without question. Theorems are facts that have been proved true using definitions, postulates and other already proved theorems. We cannot use a theorem until we learn about it.

Example 2

In a recent post about proving The Converse of the Alternate Interior Angles Theorem, I created my first flow proof. It was a short proof, much like in this example using ASA. I color coded my markings, but on the vertical angles APX and BPY I did not get the correct shade of blue.

Example 3

As you can see, it is a fairly complicated diagram for a relatively easy proof.

Given: angle CAB is congruent to angle DAE, segment AS is congruent to segments AE and angle ABC and AED are right triangles

Prove: triangle ABC is congruent to AED

Planning this proof requires you to understand your given information and the prove statement. Since the diagram has an added dimension of difficulty because extra triangle in the middle, you need to focus on the two outer right triangles ABC and AED. These are the angles that need to be proved congruent.

It is given that angles CAB and DAE are congruent and are corresponding angles in each triangle. Also, the corresponding sides AB and AE are congruent. Finally, there is information relating to the corresponding angles ABC and AED. They are both right angles. Angles ABC and AED are the angle that gives Angle-Side-Angle in each triangle. Since all right angles are congruent, angles ABC and AED are congruent. Therefore, both triangles ABC and AED are congruent by ASA. QED

Example 4

Given: segments NM and NP are congruent and angles M and P are congruent

Prove: triangles NML and NPO congruent

There are many ways to prove the triangles congruent in this diagram, but I like to produce the most concise proof possible. Hopefully you can see the ASA relationship that will be able to be proved from this set up. Angles LNM and ONP can be proved congruent to establish the ASA congruence in these two triangles.

Since it is given that segments NM and NP are congruent and angles M and P are congruent, a ASA congruence can be established with triangles NML and NPO. Angle LNM and angle ONP are congruent because they are vertical are congruent. Finally, it can be stated that triangles NML and NPO congruent because of the Angle-Side-Angle Theorem. QED

As usual, I hope this has been helpful. Be sure to check back for more geometry.